Editing SSCpt1/Virial/FormFactors

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=Structural Form Factors (Pt 1)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" bgcolor="lightblue" width="33%"><br />[[SSCpt1/Virial/FormFactors|Part I:&nbsp; Synopsis]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue" width="33%"><br />[[SSCpt1/Virial/FormFactors/Pt2|Part II:&nbsp; n = 5 Polytrope]]
&nbsp;
  </td>
  <td align="center" bgcolor="lightblue"><br />[[SSCpt1/Virial/FormFactors/Pt3|Part III:&nbsp; n = 1 Polytrope]]
&nbsp;
  </td>
</tr>
</table>


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<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|<b>Structural<br />Form<br />Factors</b>]]</font>
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As has been defined in [[SSCpt1/Virial#Structural_Form_Factors|a companion, introductory discussion]], three key dimensionless structural form factors are:

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{f}_M </math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int_0^1  3\biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x  \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x^2 dx \biggr\}  \biggl[ \frac{\rho(x)}{\rho_0}\biggr] x dx\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_0}\biggr]  x^2 dx \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>x \equiv r/R_\mathrm{limit}</math>, and the subscript "0" denotes central values.  The principal purpose of this chapter is to carry out the integrations that are required to obtain expressions for these structural form factors, at least in the few cases where they can be determined analytically.  These form-factor expressions will then be used to provide expressions for the two constants, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, that appear in the free-energy function and in the virial theorem, and to provide corresponding expressions for the normalized energies, <math>W_\mathrm{grav}/E_\mathrm{norm}</math> and <math>S_\mathrm{therm}/E_\mathrm{norm}</math>.
&nbsp;<br />

==Synopsis==

<div align="center"><b>Summary of Derived Structural Form-Factors</b></div>
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> Polytropes <math>(n \ne 5)</math>
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> Polytropes <math>(n \ne 5)</math>
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1} 
</math>
  </td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} +  (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
  </td>
</tr>
</table>

</td>
</tr>

<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> n = 1 Polytrope<br /><math>\tilde\xi \rightarrow \xi_1 = \pi</math>
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> n = 1 Polytropes<br /><math>0 < \tilde\xi < \pi</math>
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\frac{3}{\pi^2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{3^2\cdot 5}{2^2 \pi^4} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3}{2\pi^2}  
</math>
  </td>
</tr>
</table>
</td>

<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\frac{3}{\tilde\xi^3} [\sin \tilde\xi  - \tilde\xi \cos \tilde\xi  ] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{3\cdot 5}{2^3 \tilde\xi^6} \biggl[ 4\tilde\xi^2 - 3\tilde\xi \sin(2\tilde\xi) + 2\tilde\xi^2 \cos(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3}{2^2\tilde\xi^3}  \biggl[2\tilde\xi  - \sin(2\tilde\xi ) \biggr] 
</math>
  </td>
</tr>
</table>
</td>
</tr>

<tr>
<th align="center" colspan="1">
<font color="red">Isolated</font> n = 5 Polytrope
</th>
<th align="center" colspan="1">
<font color="red">Pressure-Truncated</font> n = 5 Polytropes
</th>
</tr>
<tr>
<td align="center">
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
</td>

<td align="center">

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
( 1 + \ell^2 )^{-3/2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\frac{5}{2^4} \cdot \ell^{-5}  
\biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3}{2^3} \ell^{-3}  [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ]  
</math>
  </td>
</tr>
</table>
where, &nbsp; &nbsp; <math>\ell \equiv \frac{\tilde\xi}{\sqrt{3}}</math>
</td>
</tr>
</table>

==Expectation in Context of Pressure-Truncated Polytropes==
For pressure-truncated polytropic configurations, the normalized virial theorem states that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>2 \biggl( \frac{S_\mathrm{therm}}{E_\mathrm{norm}} \biggr) + \frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} \, .</math>
  </td>
</tr>
</table>
</div>
This provides one mechanism by which the correctness of our form-factor expressions can be checked.  Specifically, having determined <math>S_\mathrm{therm}</math> and <math>W_\mathrm{grav}</math> from the derived form factors, we can see whether the sum of these energies as specified on the lefthand-side of this virial theorem expression indeed match the normalized energy term involving the external pressure, as specified on the righthand side.  In order to facilitate this "reality check" at the end of each example, below, we will use [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's detailed force-balanced solution of the equilibrium structure of embedded polytropes]] to provide an expression for the term on the righthand side of the virial theorem expression.

We begin by plugging our [[SSCpt1/Virial#Normalizations|general expression for <math>E_\mathrm{norm}</math>]] into this righthand-side term and grouping factors to facilitate insertion of Stahler's expressions.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{tot}^{(5-n)} \biggr]^{1/(n-3)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-5)/(n-3)} P_e R_\mathrm{eq}^3 
\biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)} \, . </math>
  </td>
</tr>
</table>
</div>
From [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|Stahler's equilibrium solution]], we have,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
R_\mathrm{eq}
</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\tilde\xi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{1/2} G^{-1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~~
~P_e R_\mathrm{eq}^3
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{1 + 3(1-n)/[2(n+1)]}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
M_\mathrm{limit}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2 
\biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\tilde\xi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} G^{-3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~~
K^{-n} G^3 M_\mathrm{limit}^{(5-n)}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3-3(5-n)/2} K_n^{-n +2n(5-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{{(5-n)}/2} G^{3(n-3)/2} K_n^{3n(3-n)/(n+1)} P_\mathrm{e}^{(3-n)(5-n)/[2(n+1)]} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~~
P_e R_\mathrm{eq}^3 \biggl[ K^{-n} G^3 M_\mathrm{limit}^{(5-n)} \biggr]^{1/(n-3)}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1/2} \biggr\}^{(5-n)/(n-3)} G^{3/2} K_n^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>\times ~\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^3_{\tilde\xi}
\biggl( \frac{n+1}{4\pi} \biggr)^{3/2} G^{-3/2} K_n^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{ \biggl[ \theta_n^{(n-3)/2} \xi^2 \biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr]^{(5-n)}_{\tilde\xi}
\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{(5-n)/2}  
\biggl[ \xi \theta_n^{(n-1)/2} \biggr]^{3(n-3)}_{\tilde\xi} \biggl( \frac{n+1}{4\pi} \biggr)^{3(n-3)/2} \biggr\}^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{
(n+1)^{3[(5-n)+(n-3)]/2} (4\pi)^{[(n-5)+(9-3n)]/2}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{[(n-3)(5-n) + 3(n-1)(n-3)]/2} \tilde\xi^{[2(5-n) + 3(n-3)]}
\biggr\}^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ (n+1)^{3} (4\pi)^{(2-n)}
\biggl| \frac{d\theta_n}{d\xi} \biggr|^{(5-n)}_{\tilde\xi}
(\theta_n)_{\tilde\xi}^{(n+1)(n-3)} \tilde\xi^{(n+1)}
\biggr\}^{1/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the expectation based on Stahler's equilibrium models is that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{1/(n-3)} \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{(-\theta'_n)_{\tilde\xi}}\biggr]^{(n-5)/(n-3)} 
(\theta_n)_{\tilde\xi}^{(n+1)} \tilde\xi^{(n+1)/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

As a cross-check, multiplying this expression through by <math>[(R_\mathrm{eq}/R_\mathrm{norm})(M_\mathrm{norm}/M_\mathrm{limit})^2]</math> &#8212; where the expression for <math>R_\mathrm{eq}/R_\mathrm{norm}</math> can be obtained from our [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|discussions of detailed force-balanced models]] &#8212; gives a related result that can be obtained directly from [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Detailed_Force-Balanced_Solution|Horedt's expressions]], namely,

<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>\biggl[ \frac{4\pi P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\tilde\theta^{n+1} }{(n+1)( -\tilde\theta' )^{2}}   \, .
</math>
  </td>
</tr>
</table>
</div>

==Viala and Horedt (1974) Expressions==
===Presentation===

{{ VH74full }} have provided analytic expressions for the gravitational potential energy and the internal energy &#8212; which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively &#8212; that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres.  [The same expression for <math>~\Omega</math> is also effectively provided in &sect;1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]
<div align="center">
<table border="1" align="center" cellpadding="8" width="90%">
<tr><td align="center">

<!-- 
[[Image:VialaHoredt1974.png|500px|center]]
Astronomy &amp; Astrophysics, 33: 195-202, (1974)<br />
POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />
Y. P. Viala &amp; Gp. Horedt<br />
-->
{{ VH74figure }}
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>\Omega</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- G\int_0^M \frac{MdM}{r} = \frac{16\pi^2 G \rho_0^2 \alpha^5}{(5-n)} \biggl[
\mp \xi^3 \theta^{n+1} - 3\xi^3 (\theta')^2 - 3\xi^2 \theta (\theta')
\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>U</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\gamma - 1}\int_V pdV = \frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \int_0^\xi \theta^{n+1} 4\pi \alpha^2 \xi^2 d\xi
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1}  \cdot
\frac{4\pi \alpha^2(n+1)}{(5-n)}
\biggl[
\frac{2\xi^3 \theta^{n+1}}{n+1} \pm \xi^3 (\theta')^2 \pm \xi^2 \theta (\theta')
\biggr]_0^\xi \, .
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
(the superior sign holds if <math>-1 < n < \infty</math>, the inferior if <math>-\infty < n < -1</math>)
  </td>
</tr>
</table>

</td></tr>
<tr>
  <td align="left">
A couple of key equations drawn directly from {{ VH74 }} have been shown here.  As its title indicates, the paper includes discussion of &#8212; and accompanying equation derivations for &#8212; equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries:  planar sheets, axisymmetric cylinders, and spheres.  We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only.  These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.
  </td>
</tr>
</table>
</div>
Rewriting these two expressions to accommodate our parameter notations &#8212; recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> &#8212; we have from {{ VH74 }},

<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>\biggl[ W_\mathrm{grav} \biggr]_\mathrm{VH74}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl[ \mathfrak{S}_\mathrm{A} \biggr]_\mathrm{VH74}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{n(4\pi)^2}{3(5-n)} \cdot G \rho_c^2 a_n^5 
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

===First Reality Check===

As a quick reality check, let's see whether, when appropriately added together, these two energies satisfy the scalar virial theorem for isolated polytropes.

<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>\biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
W_\mathrm{grav} + \frac{3}{n} \mathfrak{S}_A</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>+
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5 
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\frac{6}{(n+1)}  - 1 \biggr] \tilde\xi^3 \tilde\theta^{n+1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(4\pi)^2}{(n+1)} \cdot G \rho_c^2 a_n^5 \tilde\xi^3 \tilde\theta^{n+1} \, . 
</math>
  </td>
</tr>
</table>
</div>
For ''isolated polytropes'', <math>\tilde\theta \rightarrow 0</math>, so this sum of terms goes to zero, as it should if the system is in virial equilibrium.

===Renormalization===
Both of the energy-term expressions derived by {{ VH74 }} are written in terms of <math>\rho_c</math> and 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>a_\mathrm{n}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{(n+1)K_n}{4\pi G} \cdot \rho_c^{(1-n)/n}\biggr]^{1/2} 
</math>
  </td>
</tr>
</table>
</div>
&#8212; that is, effectively in terms of <math>\rho_c</math> and {{ Template:Math/MP_PolytropicConstant}} &#8212; whereas, in the context of our discussions, we would prefer to express them in terms of [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Adopted_Normalizations|our generally adopted energy normalization]], 
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ K_n^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .</math>
  </td>
</tr>
</table>
</div>
In order to accomplish this, we need to replace the central density with the total mass of an ''isolated polytrope'', <math>M_\mathrm{tot}</math>, whose generic expression is (see, for example, equation 69 of Chandrasekhar),
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, we have,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>E_\mathrm{norm}^{n-3}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
K_n^n G^{-3}\biggl\{ (4\pi)^{-1/2} \biggl[ \frac{(n+1)K_n}{G} \biggr]^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr\}^{n-5} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} \rho_c^{(3-n)/2n} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5} 
K_n^{[2n + 3(n-5)]/2} G^{[-6-3(n-5)]/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{n-5} 
\rho_c^{(n-3)(5-n)/2n} K_n^{5(n-3)/2} G^{-3(n-3)/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} 
\rho_c^{(5-n)/2n} K_n^{5/2} G^{-3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\rho_c^{[ - 4n +(5-n)]/2n} \biggl( \frac{K_n}{G}\biggr)^{5/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{K_n}{G} \cdot \rho_c^{(1-n)/n}\biggr]^{5/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(n-5)/(n-3)} G\rho_c^2
\biggl[ \frac{4\pi}{(n+1)} \cdot a_n^2 \biggr]^{5/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~(4\pi)^2 G\rho_c^2 a_n^5</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm} (4\pi)^2 
\biggl[ (4\pi)^{-1/2} (n+1)^{3/2} (-\tilde\xi^2 \tilde\theta^')_{\xi_1} \biggr]^{(5-n)/(n-3)}  
\biggl[ \frac{(n+1)}{4\pi} \biggr]^{5/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm} (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)/(n-3)}  
(4\pi)^{[-(n-3)-(5-n)]/2(n-3)} (n+1)^{[3(5-n)+5(n-3)]/2(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)}  \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
So, employing our preferred normalization, the {{ VH74 }} expressions become,

<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>\biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-
\frac{1}{(5-n)} 
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)}  \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl[ \frac{\mathfrak{S}_\mathrm{A}}{E_\mathrm{norm}} \biggr]_\mathrm{VH74}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{n}{3(5-n)} 
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr]
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)}  \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

===Second Reality Check===
If we now renormalize the sum of energy terms discussed in our [[SSCpt1/Virial/FormFactors#First_Reality_Check|first reality check, above]], we have,

<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>
\frac{1}{E_\mathrm{norm}} \biggl[ W_\mathrm{grav} + 2S_\mathrm{therm} \biggr]_\mathrm{VH74}
= \frac{4\pi P_e R_\mathrm{eq}^3}{E_\mathrm{norm}}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(n+1)^{-1} \tilde\xi^3 \tilde\theta^{n+1} \biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)}  \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>

(This may or may not be useful!)

===Implication for Structural Form Factors===
On the other hand, our expressions for these two [[SSCpt1/Virial#Structural_Form_Factors|normalized energy components written in terms of the structural form factors]] are,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
  </td>
</tr>
</table>
</div>
where, in equilibrium (see [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|here]] and [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|here]] for details),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \biggl\{ \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}}\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
\biggl\{ \biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-1)/(n-3)}
\biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_M </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>
Hence, we deduce that,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math> - \frac{5}{3} \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]
\chi_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \cdot \tilde\mathfrak{f}^2_M 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{5}{3} \biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggr\}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{[(n-1)-2(n-3)]/(n-3)} 
\cdot \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr)^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{[(1-n)+2(n-3)]/(n-3)} \tilde\xi^{[-(n-3)+2(1-n)]/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} 
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{(n-5)/(n-3)} \tilde\xi^{(5-3n)/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
If we now adopt the {{ VH74hereafter }} expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl\{~~~\biggr\}_\mathrm{VH74}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(5-n)} 
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)}  \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} 
\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} 
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(5-n)} 
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
(-\tilde\xi^2 \tilde\theta^')^{(5-n)/(n-3)}  \, .
</math>
  </td>
</tr>
</table>
</div>
Therefore, 
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\biggl[ \tilde\mathfrak{f}_W \biggr]_\mathrm{VH74}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{3\cdot 5}{(5-n)} \biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
\tilde\xi^{-5}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
\, .
</math>
  </td>
</tr>
</table>
</div>

Now, from [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#PTtable|our earlier work]] we deduced that <math>\tilde\mathfrak{f}_A</math> is related to <math>\tilde\mathfrak{f}_W</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\tilde\theta^{n+1} + \tilde\mathfrak{f}_W\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \, .</math>
  </td>
</tr>
</table>
</div>
Hence, we now have,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\biggl[ \tilde\mathfrak{f}_A \biggr]_\mathrm{VH74}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\tilde\theta^{n+1} + \frac{(n+1)}{(5-n)} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} +  (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
\, .
</math>
  </td>
</tr>
</table>
</div>

Building on the work of {{ VH74hereafter }}, we have, quite generally,

<div align="center" id="PTtable">
<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1} 
</math>
  </td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} +  (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
  </td>
</tr>
</table>

</td>
</tr>
<tr>
  <td align="left" colspan="2">
We should point out that {{ LRS93bfull }} define a different set of dimensionless structure factors for ''isolated'' polytropic spheres  &#8212; <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10)  is used in the determination of the gravitational potential energy.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>k_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{n(n+1)}{5-n} \biggr] \xi_1|\theta^'_1|</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>k_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{3}{5-n} \biggl[ \frac{4\pi |\theta^'_1|}{\xi_1} \biggr]^{1 / 3} </math>
  </td>
</tr>
</table>
</div>
Note that these are defined in the context of energy expressions wherein the central density, rather than the configuration's radius, serves as the principal parameter.  We note, as well, that for rotating configurations they define two additional dimensionless structure factors &#8212; <math>k_3</math> (their equation 3.17) is used in the determination of the rotational kinetic energy; and <math>\kappa_n</math> (their equation 3.14; also equation 7.4.9 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>])  is used in the determination of the moment of inertia.
  </td>
</tr>
</table>
</div>

The singularity that arises when <math>n = 5</math> leads us to suspect that these general expressions fail in that one specific case.  Fortunately, as [[#Summary_.28n.3D5.29|we have shown in an accompanying discussion]], <math>\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, as well as <math>\mathfrak{f}_M</math>, can be determined by direct integration in this single case.

===Related Discussions===
* See [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Model_Sequences|our plot of, what Kimura (1981b) would refer to as, several <math>M_1</math> sequences]]

=See Also=
<ul>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>


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